A proof that is only based on visual elements, without any comments.
An arithmetic identity can be demonstrated by a picture showing a self-evident equality between numerical quantities. The above figure shows that the difference between
the pentagonal number and triangular number .
Of course, the situation depicted is a particular case of the formula (here it corresponds
to
Another form of proof without words frequently used in elementary geometry is the
dissection proof .
See also Arithmetic-Logarithmic-Geometric Mean Inequality ,
Dissection Proof ,
Gabriel's
Staircase ,
Odd Number Theorem ,
Proof ,
Prosthaphaeresis Formulas ,
Trigonometric
Addition Formulas
This entry contributed by Margherita
Barile
Explore with Wolfram|Alpha
References Ayoub, A. B. "Proof without Words: Arithmetic Mean-Geometric Mean Inequality." Math. Computer Educ. 31 , 191,
1997. Bogomolny, A. "Proofs without Words." https://cut-the-knot.org/ctk/pww.shtml . Chilaka,
J. O. "Proof without Words." Math. Computer Educ. 30 ,
312, 1996. Chilaka, J. O. "Proof without Words: A Combinatorial
Identity for Math. Computer Educ. 35 , 43, 2001. Harvard University Instructional
Computing Group. "Proof without Words: Sum of Squared Integers." http://icg.harvard.edu/~gov3009/spring02/sumsq.pdf . Morey,
J. "Proof without Words." http://www.math.ubc.ca/~morey/talk/proofwowords.html . Nelsen,
R. B. Proofs
Without Words: Exercises in Visual Thinking. Nelsen, R. B. "Proof without Words: Sums of Integers
as Sums of Cubes." Math. Mag. 71 , 65, 1998. Nelsen,
R. B. Proofs
Without Words II: More Exercises in Visual Thinking. Sher, D. "Proof Without Words: Math. Computer Educ. 31 ,
190, 1997. Sher, D. "Proof Without Words: Math. Computer
Educ. 32 , 51, 1998. Wilson, J. http://jwilson.coe.uga.edu/emt725/AMGM/AMGM.1.html . Wise,
D. S. "Proof without Words: A Generalization from Pythagoras." Math.
Mag. 71 , 64, 1998. Referenced on Wolfram|Alpha Proof without Words
Cite this as:
Weisstein, Eric W. , with contributions by Margherita Barile . "Proof without Words." From MathWorld https://mathworld.wolfram.com/ProofwithoutWords.html
Subject classifications
th
pentagonal number and 
is equal to three times the 
th triangular number.
Of course, the situation depicted is a particular case of the formula (here it corresponds
to 
),
but it is presented in a way that can be immediately generalized.
."
Math. Computer Educ. 35, 43, 2001.Harvard University Instructional
Computing Group. "Proof without Words: Sum of Squared Integers." 
. Math. Computer Educ. 31,
190, 1997.Sher, D. "Proof Without Words: 
." Math. Computer
Educ. 32, 51, 1998.Wilson, J.